I used to visit the link above a couple of times. Each time, the initial direction of spinning was different. The reason is that the image is, in reality, a 2D image shifting back and forth. According to this source, our brain has not evolved to deal with 2D images, so it takes clues from the image to reconstruct a 3D model. This happens in the Visual Cortex. It will scramble along and try to make sense of what it sees. Either direction of rotation works, so whatever is found first, it will stick. Unless you refocus and give your visual cortex a chance to come to a different conclusion.
Here a quote from the source:
This news article, like many others, ignores the true source of this optical illusion and instead claims it is a quick test to see if you use more of your right brain or left brain. This is utter nonsense, but the “right-brain/left brain” thing is in the public consciousness and won’t be going away anytime soon. Sure, we have two hemispheres that operate fine independently and have different abilities, but they are massively interconnected and work together as a seamless whole (providing you have never had surgery to cut your corpus callosum).
We also do have hemispheric dominance, but that determines mostly your handedness and the probability of language being on the right or the left. There is also often asymmetry for memory, with some being right or left hemisphere dominant. But none of this means that your personality or abilities are more right brain or left brain. That much is nonsense.
Further, how your visual cortex constructs this optical illusion says nothing about your hemispheric dominance, and is absolutely not a quick personality profile.
EDIT: I just found this image: It allows immediate switching by looking at either the left or the right instance.